Modeling Bond Yields in Finance and Macroeconomics

FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES

Modeling Bond Yields in Finance and Macroeconomics

Francis X Diebold University of Pennsylvania

Monika Piazzesi University of Chicago

and Glenn D. Rudebusch Federal Reserve Bank of San Francisco

Working Paper 2005-04

The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.

Modeling Bond Yields in Finance and Macroeconomics Francis X. Diebold, Monika Piazzesi, and Glenn D. Rudebusch*

January 2005

Francis X. Diebold, Department of Economics, University of Pennsylvania, Philadelphia, PA 19104 Phone: 215-898-1507, Email: fdiebold@sasupenn.edu Monika Piazzesi, Graduate School of Business, University of Chicago, Chicago IL 60637 Phone: 773-834-3199, Email: monika.piazzesi@gsb.uchicago.edu Glenn D. Rudebusch, Economic Research, Federal Reserve Bank of San Francisco, 101 Market Street, San Francisco CA 94105 Phone: 415-974-3174, Email: glenn.rudebusch@sf.

Abstract: From a macroeconomic perspective, the short-term interest rate is a policy instrument under the direct control of the central bank. From a finance perspective, long rates are risk-adjusted averages of expected future short rates. Thus, as illustrated by much recent research, a joint macro-finance modeling strategy will provide the most comprehensive understanding of the term structure of interest rates. We discuss various questions that arise in this research, and we also present a new examination of the relationship between two prominent dynamic, latent factor models in this literature: the Nelson-Siegel and affine no-arbitrage term structure models.

*The views expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of San Francisco. We thank our colleagues and, in particular, our many co-authors.

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From a macroeconomic perspective, the short-term interest rate is a policy instrument under the direct control of the central bank, which adjusts the rate to achieve its economic stabilization goals. From a finance perspective, the short rate is a fundamental building block for yields of other maturities, which are just risk-adjusted averages of expected future short rates. Thus, as illustrated by much recent research, a joint macro-finance modeling strategy will provide the most comprehensive understanding of the term structure of interest rates. In this paper, we discuss some salient questions that arise in this research, and we also present a new examination of the relationship between two prominent dynamic, latent factor models in this literature: the Nelson-Siegel and affine no-arbitrage term structure models. I. Questions about Modeling Yields

(1) Why use factor models for bond yields? The first problem faced in term structure modeling is how to summarize the price information at any point in time for the large number of nominal bonds that are traded. In fact, since only a small number of sources of systematic risk appear to underlie the pricing of the myriad of tradable financial assets, nearly all bond price information can be summarized with just a few constructed variables or factors. Therefore, yield curve models almost invariably employ a structure that consists of a small set of factors and the associated factor loadings that relate yields of different maturities to those factors. Besides providing a useful compression of information, a factor structure is also consistent with the celebrated "parsimony principle," the broad insight that imposing restrictions?even those that are false and may degrade in-sample fit?often helps both to avoid data mining and to produce good forecasting models. For example, an unrestricted

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Vector Autoregression (VAR) provides a very general linear model of yields, but the large number of estimated coefficients renders it of dubious value for prediction (Diebold and Calin Li, 2005). Parsimony is also a consideration for determining the number of factors needed, along with the demands of the precise application. For example, to capture the time series variation in yields, one or two factors may suffice since the first two principal components account for almost all (99%) of the variation in yields. Also, for forecasting yields, using just a few factors may often provide the greatest accuracy. However, more than two factors will invariably be needed in order to obtain a close fit to the entire yield curve at any point in time, say, for pricing derivatives.

(2) How should bond yield factors and factor loadings be constructed? There are a variety of methods employed in the literature. One general approach places structure only on the estimated factors. For example, the factors could be the first few principal components, which are restricted to be mutually orthogonal, while the loadings are relatively unrestricted. Indeed, the first three principal components typically closely match simple empirical proxies for level (e.g., the long rate), slope (e.g., a long minus short rate), and curvature (e.g., a midmaturity rate minus a short and long rate average). A second approach, which is popular among market and central bank practitioners, is a fitted Nelson-Siegel curve (introduced in Charles Nelson and Andrew Siegel, 1987). As described by Diebold and Li (2005), this representation is effectively a dynamic three-factor model of level, slope, and curvature. However, the Nelson-Siegel factors are unobserved, or latent, which allows for measurement error, and the associated loadings have plausible economic restrictions (forward rates are always positive, and the discount factor approaches zero as maturity increases). A third approach is the no-arbitrage dynamic latent factor model, which is the model of choice

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in finance. The most common subclass of these models postulates flexible linear or affine forms for the latent factors and their loadings along with restrictions that rule out arbitrage strategies involving various bonds.

(3) How should macroeconomic variables be combined with yield factors? Both the Nelson-Siegel and affine no-arbitrage dynamic latent factor models provide useful statistical descriptions of the yield curve, but they offer little insight into the nature of the underlying economic forces that drive its movements. To shed some light on the fundamental determinants of interest rates, researchers have begun to incorporate macroeconomic variables into these yield curve models.

For example, Diebold, Rudebusch, and S. Boragan Aruoba (2005) provide a macroeconomic interpretation of the Nelson-Siegel representation by combining it with VAR dynamics for the macroeconomy. Their maximum likelihood estimation approach extracts three latent factors (essentially level, slope, and curvature) from a set of 17 yields on U.S. Treasury securities and simultaneously relates these factors to three observable macroeconomic variables (specifically, real activity, inflation, and a monetary policy instrument).

The role of macroeconomic variables in a no-arbitrage affine model is explored by several papers. In Piazzesi (2005), the key observable factor is the Federal Reserve's interest rate target. The target follows a step function or pure jump process, with jump probabilities that depend on the schedule of policy meetings and three latent factors, which also affect risk premiums. The short rate is modeled as the sum of the target and short-lived deviations from target. The model is estimated with high-frequency data and provides a new identification scheme for monetary policy. The empirical results show that relative to standard latent factor models using macroeconomic information can substantially lower pricing errors. In

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